FINDING AREA, PERIMETER, AND CIRCUMFERENCE
Area, perimeter, and circumference are all measures of two-dimensional shapes. These
are things you can think of as flat: a football field, a piece of paper, or a pizza. You're
probably not interested in how high they are, but you might want to know their:
● Perimeter or Circumference. This is the total length of a shape's outline. If you
built a fence around its edge, how long would that fence be? If you walked
around the edges of this area, how far would you have gone? The length of a
straight-sided shape's outline is called its perimeter, and the length of a circle's
outline is called its circumference.
● Area. This is the total amount of space inside a shape's outline. If you wanted to
paint a wall or irrigate a circular field, how much space would you have to cover?
Triangles
1. The perimeter of any triangle is the sum of its sides: a + b + c
perimeter = a + b + c
c
a
5
3
b
P=a+b+c
P=3+5+4
P = 12
4
2. The area of any triangle is half its base times its height.
area = 1/2 bh
h
3
b
4
A = 1/2 bh
A = 1/2 * 3 * 4
A = 1/2 * 12
A=6
It doesn't matter which of the triangle's short legs is the "base" and which is the
"height": you get the same solution either way.
h
A = 1/2 bh
A = 1/2 * 4 * 3
A=2*3
A=6
4
b
3
Squares
1. A square is a kind of rectangle, and the perimeter of any rectangle is the sum of its
four sides. Since all sides of a square are the same,
perimeter = 4s
s
P = 4s
P=4*3
P = 12
3
s
3
2. The area of a square is equal to any one of its sides times any other: s * s . Since
that's the same as s squared,
area =
s2
s
A = s2
A = 32
A=3*3
A=9
3
s
3
Rectangles
1. The perimeter of a rectangle is the sum of its four sides. Since a rectangle has two
equal short sides (width, w ) and two equal long sides (length, l ),
perimeter = 2l + 2w
w
3
l
7
P = 2l + 2w
P = (2 * 7) + (2 * 3)
P = 14 + 6
P = 20
2. The area of a rectangle is equal to its length times its width.
area = l * w
w
A=l*w
A=3*7
A = 21
3
l
7
Parallelograms
Like squares and rectangles, parallelograms are quadrilaterals: they have four sides
and four interior angles. In a parallelogram those angles are not right angles, but
the opposite sides must still be parallel to each other.
1. The perimeter of a parallelogram is the sum of its four sides. Since a parallelogram
has two equal short sides (width, w ) and two equal long sides (length, l ),
perimeter = 2l + 2w
w
P = 2l + 2w
P = (2*5) + (2 * 4)
P = 10 +8
P = 18
4
l
5
2. The area of a parallelogram is equal to its base (another name for length) times
its height. Its height is not the same as its width: height is measured by a vertical line
perpendicular (at right angles to) the base.
area = b * h
h
A=b*h
A=3*5
A = 15
3
b
5
Trapezoids
A trapezoid is also a quadrilateral: it has four sides, but only two are parallel.
1. The perimeter of a trapezoid is the sum of its four sides.
perimeter = a + b + c + d
b
a
2
c
4
3
d
P=a+b+c+d
P=2+3+4+5
P = 14
5
2. To find the area of a trapezoid, we use its two bases and its height:
A = 1/2 (b 1 + b 2 ) * (h)
area = 1/2 (b1 + b2) (h)
b1
h
2
3
b2
5
A = 1/2 (2 + 5) * 3
A = 1/2 * 7 * 3
A = 1/2 * 21
A = 10.5
Circles
To find a circle's circumference or area, you first need to know either its
radius:
r , the distance from its center to any point on its outer edge, or its
diameter: d , the length of a straight line through the circle's center that touches
any two points on the outer edge.
A circle's radius is always exactly half its diameter.
r
d
1. The circumference of any circle equals two times its radius multiplied by pi
(π, approximately 3.14). We can also say it equals pi times its diameter.
circumference = 2 π r
r
OR
πd
3
C=2 π*3
C=6* π
C ≈ 18.84
Because 3.14 is only an approximate value for pi, we replace the "equals" sign (=)
with the "approximately equals" sign ( ≈). For accuracy, some teachers prefer
to use the symbol: the circumference of this circle is 6 π .
2. To find the area of a circle, square its radius and multiply the result by pi .
area =
πr
2
r
3
A = πr 2
A = 32 * π
A = (3 * 3) * π
A=9* π
A = 9π or ≈ 28.26
Created Fall 2012
Pocatello
REND 327
208-282-3662
ISU Math Center
Student Success Center
www.isu.edu/success/math
Idaho Falls
CHE 220
208-282-7925